$$F(X,Y) = 0$$
A rational point is $(x,y) \in \mathbf{Q}\times\mathbf{Q}$ such that $F(x,y) = 0$.
Definition: An integer $n$ is a congruent number if $n$ is the area of a right triangle with rational side lengths.
Open Problem: Give an algorithm to decide whether or not an integer $n$ is a congruent number.
This is a 1000-year old open problem, perhaps the oldest open problem in mathematics.
{{{id=3| T = line([(0,0), (3,0), (3,4), (0,0)],rgbcolor='black',thickness=2) lbl = text("3",(1.5,-.5),fontsize=28) + text("4",(3.2,1.5),fontsize=28) lbl += text("5",(1.5,2.5),fontsize=28) lbl += text("Area $n = 6$", (2.1,1.2), fontsize=28, rgbcolor='red') show(T+lbl, axes=False) /// }}}Theorem: The Birch and Swinnerton-Dyer Conjecture provides a solution to the congruent number problem.
Proof: Suppose $n$ is a positive integer. Consider the cubic curve $y^2 = x^3 - n^2 x$. Using algebra (see next slide), one sees that this cubic curve has infinitely many rational points if and only if there are rationals $a,b,c$ such that $n=ab/2$ and $a^2 + b^2 = c^2$. The Birch and Swinnerton-Dyer conjecture gives an algorithm to decide whether or not any cubic curve has infinitely many solutions.
{{{id=13| /// }}}
In fact, there is a bijection between
$$
A = \left\{(a,b,c) \in \mathbf{Q}^3 \,:\, \frac{ab}{2} = n,\, a^2 + b^2 = c^2\right\}
$$
and
$$
B = \left\{(x,y) \in \mathbf{Q}^2 \,:\, y^2 = x^3 - n^2 x, \,\,\text{with } y \neq 0\right\}
$$
given explicitly by the maps
$$
f(a,b,c) = \left(-\frac{nb}{a+c},\,\, \frac{2n^2}{a+c}\right)
$$
and
$$
g(x,y) = \left(\frac{n^2-x^2}{y},\,\,-\frac{2xn}{y},\,\, \frac{n^2+x^2}{y}\right).
$$